| L(s) = 1 | − 5-s + 3·7-s − 3·9-s + 5·11-s − 5·13-s − 4·17-s + 8·19-s + 23-s + 5·25-s − 9·29-s + 31-s − 3·35-s + 12·37-s − 3·41-s + 43-s + 3·45-s + 3·47-s + 7·49-s − 4·53-s − 5·55-s + 11·59-s + 7·61-s − 9·63-s + 5·65-s − 67-s + 8·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 9-s + 1.50·11-s − 1.38·13-s − 0.970·17-s + 1.83·19-s + 0.208·23-s + 25-s − 1.67·29-s + 0.179·31-s − 0.507·35-s + 1.97·37-s − 0.468·41-s + 0.152·43-s + 0.447·45-s + 0.437·47-s + 49-s − 0.549·53-s − 0.674·55-s + 1.43·59-s + 0.896·61-s − 1.13·63-s + 0.620·65-s − 0.122·67-s + 0.949·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780337402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780337402\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.35189272500280184237064208959, −10.72893673782673732821568918200, −9.819470853695833479821261566070, −9.655315308619979168593484329107, −9.130423303298720833791456955605, −8.862585617961254780740691537800, −8.166221704813439176829033591306, −7.987446161744853082248826737896, −7.18959979732730806452030239329, −7.15647178722108643884658545671, −6.53189690748097713866512049406, −5.76001652068867621954264549495, −5.36825962588137218970405058696, −4.95399701410627557008623306328, −4.25896136730717251019945064395, −3.96432099671341392538325693975, −3.06682822576520482311985729038, −2.56771223640190494402709759702, −1.73617124899031196051061078570, −0.796048099372776484310285925970,
0.796048099372776484310285925970, 1.73617124899031196051061078570, 2.56771223640190494402709759702, 3.06682822576520482311985729038, 3.96432099671341392538325693975, 4.25896136730717251019945064395, 4.95399701410627557008623306328, 5.36825962588137218970405058696, 5.76001652068867621954264549495, 6.53189690748097713866512049406, 7.15647178722108643884658545671, 7.18959979732730806452030239329, 7.987446161744853082248826737896, 8.166221704813439176829033591306, 8.862585617961254780740691537800, 9.130423303298720833791456955605, 9.655315308619979168593484329107, 9.819470853695833479821261566070, 10.72893673782673732821568918200, 11.35189272500280184237064208959
